SOLUTION: In describing her plot of land to the surveyor Jemma provided the following details: One side is lined by a 40m fence. Another side runs parallel to the road for 22meters. Those

Question 1180768: In describing her plot of land to the surveyor Jemma provided the following details: One side is lined by a 40m fence. Another side runs parallel to the road for 22meters. Those two sides create a 30 degree angle between them. The inspector says this information can create 2 different triangular plots of land. Show work to calculate the remaining angles and the third side of the triangle for both possible plots of land. Round your measurements to the tenths place.
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

The inspector is wrong....

The given information is the lengths of two sides and the measure of the angle between them. That information determines a unique triangle.

Use the law of cosines to find the measure of the third side.
Then use the law of sines to find the measure of one of the other angles.
Then use the fact that the sum of the angle measures is 180 degrees to find the third angle. (Or you could use the law of sines again to find that third angle.)

Answer by ikleyn(52802)   (Show Source): You can put this solution on YOUR website!
.

It is true that in Geometry there is only one triangle with two given sides and the concluded angle given.

All other triangles are CONGRUENT to this triangle.

It means that all other congruent triangles can be over-imposed using translations, rotations AND/OR mirror reflections.

So, from the Geometry point of view, all these triangles are equivalent.

            But living on the Earth surface, we can only move triangles, using
            translations and/or rotations - we CAN NOT make a MIRROR REFLECTION, in fact.

It is why in this scenario the inspector says that TWO DIFFERENT triangle plots are possible, based on given description.

If you imagine that the road is in West-East direction and the plot is on the North side of the road,
then one plot can be presented by a triangle skewed West, while another plot can be presented by a triangle skewed East.

Although the triangles are CONGRUENT and are non-distinguishable from the formal Geometry point of view,
NETHERVELESS they are different on the Earth surface and CAN NOT be overposed by translations and rotations,
since they are mirror images one to another.

At this point I complete my explanation about the different plots - on the Earth surface they are really different and
can not be overposed by translations and rotations.

Regarding calculation the third side of the triangle, use the Cosine Law.

Regarding finding two other angles of triangles, use the Sine Law.

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Have a nice day and enjoy my explanation (!)

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